Thread: Two questions: hypothesis test & how to win the car?

1) Generally, 95% confidence indicates that the mean effect of the drug is larger than the mean effect of the placebo 95% of the time over many repetitions.

Assumptions are: what distribution is used to calculate confidence, the sample size, width of interval, is the control group independent of the drug group or is it a longitudinal study, is there any regression toward the mean, etc.

2) This question could be difficult or easy. I would answer this question assuming that the other players pick a number 1, 2, or 3 at random.

Assuming random picks, there are 27 combinations of (a, b, c) where a, b, c are the players and a, b, and c can take on 1, 2, or 3. If you list all combinations, you will win 8/27 times if you pick completely at random, but 4 of those wins will be if you pick the number 1. As, there are 9 combinations of picks when you choose 1, you win 4/9 of the time if you pick 1, 2/9 each player 2 and player 3 will win, and the company will win if you all pick 1.

Thus, I would pick 1. Of course, all players, if intelligent would know that 1 is the best, so there could be some psychological game theory considerations, but I don't think that is what the question is asking.

Hope this helps.

[This question took 10 minutes to answer]

Can anyone shed some light on the below:

1. Consider a set with N distinct members, and a function f defined on Q that takes the values 0, 1 such that (1/N)*Sum(over xEQ) of f(x) = p. For a subset S of Q of size n, define the sample proportion

p := p(S) = (1/N)*Sum(over xES) of f(x)

If each subset of size n is chosen with equal probability, calculate the expectation and
standard deviation of the random variable p.

2.
(a) Let X-N(0, 1) be a normally distributed random variable with mean 0 and
variance 1. Suppose that x E R, x > 0. Find upper and lower bounds for the conditional expectation

E(X | X >x)

(b) Now suppose that X has a power law distribution, P(X >x) = a*(x)^-b, for x>x0>0, and some a> 0, b> 1. Calculate the conditional expectation
E(X|X>x), x >x0

Many thanks in advance.